Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| spielplatz:mathematische-formeln-anleitung [2017/12/18 14:36] – angelegt wi | spielplatz:mathematische-formeln-anleitung [2023/04/18 09:12] (aktuell) – [Mathematische Formeln in Dokuwiki] wi | ||
|---|---|---|---|
| Zeile 1: | Zeile 1: | ||
| + | ====== Mathematische Formeln in Dokuwiki ====== | ||
| + | < | ||
| + | Zusammengestellt von M. Wilfling, HTBLA Kaindorf | ||
| + | $ \frac{1}{x} $ | ||
| + | Wer möchte seine Wiki Seiten, die mathematische Formeln enthalten, professionell gestalten? | ||
| + | |||
| + | Falls in Wiki Seiten mathematische Formeln vorkommen, erfolgt dies auf Mediawiki oder z.B. wie hier in Dokuwiki mit der Latex-Syntax. Latex ist ist ein Textverarbeitungssystem, | ||
| + | |||
| + | Eine große Hemmschwelle beim Einsatz von Latex ist die Tatsache, dass Latex nicht nach dem Prinzip '' | ||
| + | |||
| + | ===== Befehle zur Formeleingabe ===== | ||
| + | |||
| + | <WRAP tip> | ||
| + | **Keine Angst vor Befehlen!** | ||
| + | |||
| + | Alle unten stehenden Beispiele können leicht kopiert, modifiziert und in eigenen Wikis eingebaut werden. | ||
| + | </ | ||
| + | |||
| + | In modernen Wikis (mit installiertem '' | ||
| + | |||
| + | Bitte unbedingt von oben nach unten lesen. Es ist für alle etwas dabei: Einsteiger eher weiter oben suchen, nach unten eher für Fortgeschrittene! | ||
| + | ===== Mathemmatische Umgebungen im Wiki Text ===== | ||
| + | |||
| + | Mit der folgenden Syntax erkennt das installierte Plugin den Code für mathematische Befehle, der dann kompiliert wird: | ||
| + | ^ Eingabe ^ Auswirkung | | ||
| + | | '' | ||
| + | | '' | ||
| + | | '' | ||
| + | | '' | ||
| + | | '' | ||
| + | | '' | ||
| + | | '' | ||
| + | |||
| + | ===== Beispiele ===== | ||
| + | |||
| + | ==== Inline Formel ==== | ||
| + | Dieses ist eine wichtige Formel: $c = a + b$ | ||
| + | Dieses ist eine wichtige Formel: $c = a + b$ | ||
| + | Für alle Zahlen $a_1, \dots, a_n$ gilt | ||
| + | Für alle Zahlen $a_1, \dots, a_n$ gilt | ||
| + | Seien $a$ und $b$ die Katheten und $c$ die Hypotenuse, dann gilt | ||
| + | $c=\sqrt{a^2+b^2}$ (Lehrsatz des Pythagoras). | ||
| + | Seien $a$ und $b$ die Katheten und $c$ die Hypotenuse, dann gilt | ||
| + | $c=\sqrt{a^2+b^2}$ (Lehrsatz des Pythagoras). | ||
| + | |||
| + | ==== Abgesetzte Formel ==== | ||
| + | Dieses ist eine wichtige Formel: $$y = f(x) = k~x + d$$ | ||
| + | Dieses ist eine wichtige Formel: $$y = f(x) = k~x + d$$ | ||
| + | |||
| + | Seien $a$ und $b$ die Katheten und $c$ die Hypotenuse, dann gilt $$c^2=a^2+b^2$$ | ||
| + | (Lehrsatz des Pythagoras). | ||
| + | Seien $a$ und $b$ die Katheten und $c$ die Hypotenuse, dann gilt $$c^2=a^2+b^2$$ | ||
| + | (Lehrsatz des Pythagoras). | ||
| + | |||
| + | ==== Zwischenraum, | ||
| + | $ab$, $a\,b$, $a\:b$, $a\;b$, $a\!b$, $a \quad b$, $a \qquad b$ | ||
| + | |||
| + | $ab$, $a\,b$, $a\:b$, $a\;b$, $a\!b$, $a \quad b$, $a \qquad b$ | ||
| + | |||
| + | ==== Ein nummerierter Formelblock ==== | ||
| + | < | ||
| + | f(x) & = & \cos x \\ | ||
| + | f’(x) & = & - \sin x \\ | ||
| + | \int_0^xf(y)\mathrm{d}y& | ||
| + | \end{eqnarray}</ | ||
| + | \begin{eqnarray} | ||
| + | f(x) & = & \cos x \\ | ||
| + | f’(x) & = & - \sin x \\ | ||
| + | \int_0^xf(y)\mathrm{d}y& | ||
| + | \end{eqnarray} | ||
| + | Das Zeichen ''&'' | ||
| + | < | ||
| + | & a+b & c+d\\ | ||
| + | & x+y & u+v\\ | ||
| + | \end{eqnarray}</ | ||
| + | \begin{eqnarray} | ||
| + | & a+b & c+d\\ | ||
| + | & x+y & u+v | ||
| + | \end{eqnarray} | ||
| + | |||
| + | ==== Feld von Formeln ohne Nummerierung ==== | ||
| + | < | ||
| + | & a+b & c+d\\ | ||
| + | & x+y & u+v\\ | ||
| + | \end{eqnarray*}</ | ||
| + | & a+b & c+d\\ | ||
| + | & x+y & u+v | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | ==== Sonderzeichen und Griechische Buchstaben ==== | ||
| + | $ \mathrm{A}\mathrm{B}\Gamma\Delta\mathrm{E}\mathrm{Z}\mathrm{H}\Theta\mathrm{I}\mathrm{K}\Lambda | ||
| + | \mathrm{M}\mathrm{N}\Xi\mathrm{O}\Pi\mathrm{P}\Sigma\mathrm{T}\Phi\mathrm{X}\mathrm{Y}\Psi\Omega$ | ||
| + | |||
| + | $ \mathrm{A}\mathrm{B}\Gamma\Delta\mathrm{E}\mathrm{Z}\mathrm{H}\Theta\mathrm{I}\mathrm{K}\Lambda \mathrm{M}\mathrm{N}\Xi\mathrm{O}\Pi\mathrm{P}\Sigma\mathrm{T}\Phi\mathrm{X}\mathrm{Y}\Psi\Omega$ | ||
| + | |||
| + | $ \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa\lambda\mu\nu\xi\mathrm{o}\pi\rho\sigma\tau\phi\chi\upsilon\psi\omega$ | ||
| + | $ \alpha\beta\gamma\delta\epsilon\zeta\eta\theta\iota\kappa\lambda\mu\nu\xi\mathrm{o}\pi\rho\sigma\tau\phi\chi\upsilon\psi\omega$ | ||
| + | $\varepsilon \quad \vartheta \quad \varpi \quad \varrho \quad \varsigma \quad \varphi$ | ||
| + | $ \varepsilon \quad \vartheta \quad \varpi \quad \varrho \quad \varsigma \quad \varphi$ | ||
| + | $\aleph\quad\Re\quad\Im\quad\partial\quad\infty\quad\forall\quad\exists\quad\neg\quad\in\quad\heartsuit$ | ||
| + | $ \aleph\quad\Re\quad\Im\quad\partial\quad\infty\quad\forall\quad\exists\quad\neg\quad\in\quad\heartsuit$ | ||
| + | $ \forall \varepsilon> | ||
| + | $ \forall \varepsilon> | ||
| + | |||
| + | Eurozeichen mit Unicode: $ \unicode{0x20AC}$ | ||
| + | Eurozeichen mit Unicode: $ \unicode{0x20AC}$ | ||
| + | ==== Klammern ==== | ||
| + | $ ( ~ \lbrack ~ \lbrace ~ [ ~ \lfloor ~ \langle ~ \{ ~ \lceil$ | ||
| + | $ ( ~ \lbrack ~ \lbrace ~ [ ~ \lfloor ~ \langle ~ \{ ~ \lceil$ | ||
| + | $ ) ~ \rbrack ~ \rbrace ~ ]~ \rfloor ~ \rangle ~ \} ~ \rceil$ | ||
| + | $ ) ~ \rbrack ~ \rbrace ~ ]~ \rfloor ~ \rangle ~ \} ~ \rceil$ | ||
| + | $ \Bigl( (x+1) (x-1)\Bigr) ^2$ | ||
| + | $ \Bigl( (x+1) (x-1)\Bigr) ^2$ | ||
| + | $ \left( (x+1) (x-1)\right) ^2$ | ||
| + | $ \left( (x+1) (x-1)\right) ^2$ | ||
| + | $ 1 + \left(\frac{1}{1-x^2}\right)$ | ||
| + | $ 1 + \left(\frac{1}{1-x^2}\right)$ | ||
| + | $ \underbrace{\overbrace{a + b + \cdots +z}^{26} + \overbrace{A + B + \cdots+Z}^{26}}_{52}$ | ||
| + | $ \underbrace{\overbrace{a + b + \cdots +z}^{26} + \overbrace{A + B + \cdots+Z}^{26}}_{52}$ | ||
| + | $ \overline{m+n} \qquad \underline{m+n}$ | ||
| + | $ \overline{m+n} \qquad \underline{m+n}$ | ||
| + | ==== Operatoren ==== | ||
| + | $x = y > z \qquad x := y \qquad x \le y \ne z $ | ||
| + | $x = y > z \qquad x := y \qquad x \le y \ne z$ | ||
| + | $x \sim y \simeq z \qquad x \equiv y \not\equiv z \qquad x \subset y \subseteq z$ | ||
| + | $x \sim y \simeq z \qquad x \equiv y \not\equiv z \qquad x \subset y \subseteq z$ | ||
| + | $x + y - z \qquad x * y / z \qquad x \times y \cdot z$ | ||
| + | $x + y - z \qquad x * y / z \qquad x \times y \cdot z$ | ||
| + | $x \circ y \bullet z \qquad x \cup y \cap z \qquad x \sqcup y \sqcap z$ | ||
| + | $x \circ y \bullet z \qquad x \cup y \cap z \qquad x \sqcup y \sqcap z$ | ||
| + | $x \vee y \wedge z\qquad x \pm y \mp z$ | ||
| + | $x \vee y \wedge z\qquad x \pm y \mp z$ | ||
| + | |||
| + | ==== Akzente ==== | ||
| + | $ \hat a \qquad \check b \qquad \tilde c \qquad \acute d \qquad \grave e$ | ||
| + | $ \hat a \qquad \check b \qquad \tilde c \qquad \acute d \qquad \grave e$ | ||
| + | $ \dot f\qquad\ddot g\qquad\breve h \qquad \bar k \qquad \vec l$ | ||
| + | $ \dot f\qquad\ddot g\qquad\breve h \qquad \bar k \qquad \vec l$ | ||
| + | $ \hat\imath \qquad \check\jmath$ | ||
| + | $ \hat\imath \qquad \check\jmath$ | ||
| + | $ \widehat x \qquad \widehat{xy} \qquad \widehat{xyz}$ | ||
| + | $ \widehat x \qquad \widehat{xy} \qquad \widehat{xyz}$ | ||
| + | $ \widetilde x \qquad \widetilde{xy} \qquad \widetilde{xyz}$ | ||
| + | $ \widetilde x \qquad \widetilde{xy} \qquad \widetilde{xyz}$ | ||
| + | ==== Vektoren ==== | ||
| + | $ \alpha \cdot(\vec x + \vec y) = \alpha \cdot \vec x + \alpha \cdot \vec y$ | ||
| + | $ \alpha \cdot(\vec x + \vec y) = \alpha \cdot \vec x + \alpha \cdot \vec y$ | ||
| + | $ \vec x \cdot (\vec y \cdot \vec z) \not=(\vec x\cdot\vec y)\cdot \vec z$ | ||
| + | $ \vec x \cdot (\vec y \cdot \vec z) \not=(\vec x\cdot\vec y)\cdot \vec z$ | ||
| + | $ \vec x\times (\vec y\times \vec z) \not= (\vec x \times \vec y) \times \vec z$ | ||
| + | $ \vec x\times (\vec y\times \vec z) \not= (\vec x \times \vec y) \times \vec z$ | ||
| + | ==== Pfeile ==== | ||
| + | $ \leftarrow \qquad \Leftarrow \qquad \leftrightarrow \qquad | ||
| + | \Leftrightarrow \qquad \uparrow\qquad\downarrow\qquad\nearrow$ | ||
| + | $ \leftarrow \qquad \Leftarrow \qquad \leftrightarrow \qquad \Leftrightarrow \qquad \uparrow\qquad\downarrow\qquad\nearrow$ | ||
| + | $\longleftarrow\qquad\leftharpoonup \qquad \mapsto \qquad \leadsto$ | ||
| + | $ \longleftarrow\qquad\leftharpoonup \qquad \mapsto \qquad \leadsto$ | ||
| + | $ (\mathcal{A} \Rightarrow \mathcal{B}) \Longleftrightarrow | ||
| + | (\lnot \mathcal{B} \Rightarrow \lnot \mathcal{A})$ | ||
| + | $ (\mathcal{A} \Rightarrow \mathcal{B}) \Longleftrightarrow (\lnot \mathcal{B} \Rightarrow \lnot \mathcal{A})$ | ||
| + | $(\mathcal{A} \Longleftrightarrow \mathcal{B}) \Longleftrightarrow | ||
| + | (\mathcal{A}\Rightarrow \mathcal{B}) \wedge (\mathcal{B}\Rightarrow\mathcal{A})$ | ||
| + | |||
| + | $ (\mathcal{A} \Longleftrightarrow \mathcal{B}) \Longleftrightarrow | ||
| + | (\mathcal{A}\Rightarrow \mathcal{B}) \wedge (\mathcal{B}\Rightarrow\mathcal{A})$ | ||
| + | |||
| + | ==== Schriftartwechsel ==== | ||
| + | $ \forall x\in\mathbf{R}: | ||
| + | $ \forall x\in\mathbf{R}: | ||
| + | < | ||
| + | \mathbf{A} \cdot \mathbf{x} & = & | ||
| + | \mathbf{y} \\ | ||
| + | \textrm{mit } \mathbf{A}& | ||
| + | && | ||
| + | \mathbf{x} & = & (x_1, | ||
| + | \textrm{ und}\\ | ||
| + | \mathbf{y} & = & (y_1, \cdots, y_m)\\ | ||
| + | \end{eqnarray*}</ | ||
| + | \begin{eqnarray*} | ||
| + | \mathbf{A} \cdot \mathbf{x} & = & | ||
| + | \mathbf{y} \\ | ||
| + | \textrm{mit } \mathbf{A}& | ||
| + | && | ||
| + | \mathbf{x} & = & (x_1, | ||
| + | \textrm{ und}\\ | ||
| + | \mathbf{y} & = & (y_1, \cdots, y_m)\\ | ||
| + | \end{eqnarray*} | ||
| + | ==== Indizes und Hochstellung ==== | ||
| + | $ \displaystyle a_i \qquad a_{\displaystyle i_j} $ | ||
| + | $ \displaystyle a_i \qquad a_{\displaystyle i_j} $ | ||
| + | $ \displaystyle a^{\displaystyle i} \qquad a^{\displaystyle i^j} $ | ||
| + | $ \displaystyle a^{\displaystyle i} \qquad a^{\displaystyle i^j} $ | ||
| + | $ _1x_2 \qquad x^3_4 \qquad a^{b^\alpha_\beta}_{c^\gamma_\delta} \qquad F^1_2 \qquad F{}^1_2$ | ||
| + | $ _1x_2 \qquad x^3_4 \qquad a^{b^\alpha_\beta}_{c^\gamma_\delta} \qquad F^1_2 \qquad F{}^1_2$ | ||
| + | Vergleiche $ \displaystyle _1x_2 = \sqrt{\left(\frac{p}{2}\right)^2-q}$ | ||
| + | mit $ _1x_2 = -\frac{p}{2}\pm \sqrt{\left(\frac{p}{2}\right)^2-q}$ ! | ||
| + | Vergleiche $ \displaystyle _1x_2 = \sqrt{\left(\frac{p}{2}\right)^2-q}$ | ||
| + | mit $ _1x_2 = -\frac{p}{2}\pm \sqrt{\left(\frac{p}{2}\right)^2-q}$ ! | ||
| + | |||
| + | ==== Brüche ==== | ||
| + | $ \displaystyle \frac{1}{2} \qquad \frac{n+1}{3}$ | ||
| + | $ \displaystyle \frac{1}{2} \qquad \frac{n+1}{3}$ | ||
| + | $ \displaystyle \frac{x+y^2}{m+1} \qquad \frac{x+y^2}{m} + 1 \qquad x + \frac{y^2}{m}+1$ | ||
| + | $ \displaystyle \frac{x+y^2}{m+1} \qquad \frac{x+y^2}{m} + 1 \qquad x + \frac{y^2}{m}+1$ | ||
| + | $x + \frac{\displaystyle y^2}{\displaystyle m+1} \qquad x + y^\frac{\displaystyle 2}{\displaystyle m+1}$ | ||
| + | $x + \frac{\displaystyle y^2}{\displaystyle m+1} \qquad x + y^\frac{\displaystyle 2}{\displaystyle m+1}$ | ||
| + | $ \displaystyle x + \frac{y^2}{m+1} \qquad x + y^\frac{2}{m+1}$ | ||
| + | $ \displaystyle x + \frac{y^2}{m+1} \qquad x + y^\frac{2}{m+1}$ | ||
| + | $ \displaystyle \frac{\frac{x}{y}}{2} \qquad \frac{x}{\frac{y}{2}}$ | ||
| + | $ \displaystyle \frac{\frac{x}{y}}{2} \qquad \frac{x}{\frac{y}{2}}$ | ||
| + | |||
| + | $x_0 + \frac{1}{x_1 + | ||
| + | \frac{1}{x_2 + | ||
| + | \frac{1}{x_3 + | ||
| + | \frac{1}{x_4}}}}$ | ||
| + | $x_0 + \frac{1}{x_1 + | ||
| + | \frac{1}{x_2 + | ||
| + | \frac{1}{x_3 + | ||
| + | \frac{1}{x_4}}}}$ | ||
| + | |||
| + | $ \displaystyle x_0 + \frac{1}{\displaystyle x_1 + | ||
| + | \frac{\strut 1}{\displaystyle x_2 + | ||
| + | \frac{\strut 1}{\displaystyle x_3 + | ||
| + | \frac{\strut 1}{\displaystyle x_4}}}}$ | ||
| + | |||
| + | $ \displaystyle x_0+\frac{1}{\displaystyle x_1 + | ||
| + | \frac{\strut 1}{\displaystyle x_2 + | ||
| + | \frac{\strut 1}{\displaystyle x_3 + | ||
| + | \frac{\strut 1}{\displaystyle x_4}}}}$ | ||
| + | ==== Wurzeln ==== | ||
| + | $ \sqrt 2 \qquad \sqrt{\displaystyle x^2-y^2} \qquad \sqrt{\alpha^2 + \beta^2 - \gamma^2}$ | ||
| + | $ \sqrt 2 \qquad \sqrt{\displaystyle x^2-y^2} \qquad \sqrt{\alpha^2 + \beta^2 - \gamma^2}$ | ||
| + | $ \sqrt[3]{2} \qquad \sqrt[n]{\sqrt\alpha + \sqrt\beta} \qquad \sqrt[n+1]{a^n + b^n}$ | ||
| + | $ \sqrt[3]{2} \qquad \sqrt[n]{\sqrt\alpha + \sqrt\beta} \qquad \sqrt[n+1]{a^n + b^n}$ | ||
| + | $ \sqrt a +\sqrt{b^2} +\sqrt c \qquad \sqrt{\mathstrut a} + \sqrt{\mathstrut b^2} + \sqrt{\mathstrut c}$ | ||
| + | $ \sqrt a +\sqrt{b^2} +\sqrt c \qquad \sqrt{\mathstrut a} + \sqrt{\mathstrut b^2} + \sqrt{\mathstrut c}$ | ||
| + | $ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1 + \sqrt{1+x}}}}}}$ | ||
| + | $ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1+ \sqrt{1 + \sqrt{1+x}}}}}}$ | ||
| + | ==== Binominalkoeffizienten ==== | ||
| + | ${n \choose 2} \qquad {n+1 \choose k} \qquad \frac{\displaystyle{n \choose k}}{\displaystyle 2}$ | ||
| + | ${n \choose 2} \qquad {n+1 \choose k} \qquad \frac{\displaystyle{n \choose k}}{\displaystyle 2}$ | ||
| + | ${x \atop a+b} \qquad {n \atop k+1}$ | ||
| + | ${x \atop a+b} \qquad {n \atop k+1}$ | ||
| + | $ \displaystyle \sum_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q}} a_{ij} b_{ji}$ | ||
| + | $ \displaystyle \sum_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q}} a_{ij} b_{ji}$ | ||
| + | ==== Limes, Ableitungen ==== | ||
| + | $ \displaystyle f\prime(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ | ||
| + | $ \displaystyle f\prime(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ | ||
| + | |||
| + | < | ||
| + | f(x) & = & \cos x \\ | ||
| + | f’(x) & = & -\sin x \\ | ||
| + | f’’(x) & = & -\cos x \\ | ||
| + | \end{eqnarray*}</ | ||
| + | \begin{eqnarray*} | ||
| + | f(x) & = & \cos x \\ | ||
| + | f’(x) & = & -\sin x \\ | ||
| + | f’’(x) & = & -\cos x \\ | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | < | ||
| + | h(x) & = & f(x) \cdot g(x)\\ | ||
| + | \frac{h(x)}{\mathrm{d}x} & = & f(x)\cdot\frac{g(x)}{\mathrm{d}x}+\frac{f(x)}{\mathrm{d}x} \cdot g(x) | ||
| + | \end{eqnarray*}</ | ||
| + | \begin{eqnarray*} | ||
| + | h(x) & = & f(x) \cdot g(x)\\ | ||
| + | \frac{h(x)}{\mathrm{d}x} & = & f(x)\cdot\frac{g(x)}{\mathrm{d}x}+\frac{f(x)}{\mathrm{d}x} \cdot g(x) | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | < | ||
| + | \mathbf{x} & = & \frac{1}{2} \mathbf{k} \cdot t^2 + \mathbf{v_0} \cdot t + \mathbf{x_0}\\ | ||
| + | \dot \mathbf{x} & = & \mathbf{k} \cdot t + \mathbf{v_0}\\ | ||
| + | \ddot \mathbf{x} & = & \mathbf{k} | ||
| + | \end{eqnarray*}</ | ||
| + | |||
| + | < | ||
| + | \mathbf{x} & = & \frac{1}{2} \mathbf{k} \cdot t^2 + \mathbf{v_0} \cdot t + x_0\\ | ||
| + | \dot{\mathbf{x}} & = & \mathbf{k} \cdot t + \mathbf{v_0}\\ | ||
| + | \ddot{\mathbf{x}} & = & \mathbf{k} | ||
| + | \end{eqnarray*}</ | ||
| + | \begin{eqnarray*} | ||
| + | \mathbf{x} & = & \frac{1}{2} \mathbf{k} \cdot t^2 + \mathbf{v_0} \cdot t + x_0\\ | ||
| + | \dot{\mathbf{x}} & = & \mathbf{k} \cdot t + \mathbf{v_0}\\ | ||
| + | \ddot{\mathbf{x}} & = & \mathbf{k} | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | < | ||
| + | z (x, y) & = & xy\\ | ||
| + | \frac{\partial z}{\partial x} & = & y \quad \textrm{und}\\ | ||
| + | \frac{\partial z}{\partial y} & = & x | ||
| + | \end{eqnarray*}</ | ||
| + | \begin{eqnarray*} | ||
| + | z (x, y) & = & xy\\ | ||
| + | \frac{\partial z}{\partial x} & = & y \quad \textrm{und}\\ | ||
| + | \frac{\partial z}{\partial y} & = & x | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | < | ||
| + | z (x, y) & = & \frac{xy}{x^2+y^2} \quad(\forall x, | ||
| + | \frac{\partial z}{\partial x} & = & \frac{y(y^2-x^2)}{(x^2+y^2)^2} \qquad \textrm{und}\\ | ||
| + | \frac{\partial z}{\partial y} & = & \frac{x(x^2-y^2)}{(x^2+y^2)^2} | ||
| + | \end{eqnarray*}</ | ||
| + | \begin{eqnarray*} | ||
| + | z (x, y) & = & \frac{xy}{x^2+y^2} \quad(\forall x, | ||
| + | \frac{\partial z}{\partial x} & = & \frac{y(y^2-x^2)}{(x^2+y^2)^2} \qquad \textrm{und}\\ | ||
| + | \frac{\partial z}{\partial y} & = & \frac{x(x^2-y^2)}{(x^2+y^2)^2} | ||
| + | \end{eqnarray*} | ||
| + | ==== Summen ==== | ||
| + | |||
| + | \sum\limits_{i=2}^{\infty} \frac{\displaystyle(-1)^i}{\displaystyle i^2}= | ||
| + | \displaystyle{\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+}~\cdots$ | ||
| + | |||
| + | $ \sum\limits_{i=2}^{\infty} \frac{\displaystyle(-1)^i}{\displaystyle i^2}=\displaystyle{\frac{1}{4}-\frac{1}{9}+\frac{1}{16}-\frac{1}{25}+}~\cdots$ | ||
| + | |||
| + | < | ||
| + | $ \sum\limits_{i=1}^p \sum\limits_{j=1}^q\sum\limits_{k=1}^r a_{ij}b_{jk}c_{ki}$ | ||
| + | < | ||
| + | \sum\limits_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q} \atop \scriptstyle 1 \le k \le r} | ||
| + | a_{ij} b_{jk} c_{ki}$ | ||
| + | </ | ||
| + | $ \sum\limits_{{\scriptstyle 1 \le i \le p \atop \scriptstyle 1 \le j \le q} \atop \scriptstyle 1 \le k \le r} a_{ij} b_{jk} c_{ki} $ | ||
| + | |||
| + | |||
| + | ==== Reihen ==== | ||
| + | < | ||
| + | \begin{eqnarray*} | ||
| + | \sum_{i=0}^{\infty}(-1)^i \frac{1}{2i+1} & = & 1 - \frac{1}{3}+\frac{1}{5}-\cdots\\ | ||
| + | & = & \frac{\pi}{4} | ||
| + | \end{eqnarray*}</ | ||
| + | |||
| + | \begin{eqnarray*} | ||
| + | \sum_{i=0}^{\infty}(-1)^i \frac{1}{2i+1} & = & 1 - \frac{1}{3}+\frac{1}{5}-\cdots\\ | ||
| + | & = & \frac{\pi}{4} | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | < | ||
| + | \begin{eqnarray*} | ||
| + | \sum_{i=1}^{\infty}(-1)^{i+1} \frac{1}{i^2} & = & 1-\frac{1}{2^2} + \frac{1}{3^2} - \cdots\\ | ||
| + | & = & \frac{\pi^2}{12} | ||
| + | \end{eqnarray*} | ||
| + | </ | ||
| + | \begin{eqnarray*} | ||
| + | \sum_{i=1}^{\infty}(-1)^{i+1} \frac{1}{i^2} & = & 1-\frac{1}{2^2} + \frac{1}{3^2} - \cdots\\ | ||
| + | & = & \frac{\pi^2}{12} | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | < | ||
| + | \begin{eqnarray*} | ||
| + | \forall x \in \mathbf{R}: | ||
| + | 1 - x + \frac{x^2}{2!} - | ||
| + | \frac{x^3}{3!} + \cdots\\ | ||
| + | & = & \sum_{i=0}^{\infty} | ||
| + | (-1)^i\frac{x^i}{i!} | ||
| + | \end{eqnarray*}</ | ||
| + | |||
| + | \begin{eqnarray*} | ||
| + | \forall x \in \mathbf{R}: | ||
| + | 1 - x + \frac{x^2}{2!} - | ||
| + | \frac{x^3}{3!} + \cdots\\ | ||
| + | & = & \sum_{i=0}^{\infty} | ||
| + | (-1)^i\frac{x^i}{i!} | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | < | ||
| + | \begin{eqnarray*} | ||
| + | \forall x \in \mathbf{R}: | ||
| + | 1 + x + \frac{x^2}{2!} + | ||
| + | \frac{x^3}{3!} + \cdots\\ | ||
| + | & | ||
| + | \end{eqnarray*}</ | ||
| + | \begin{eqnarray*} | ||
| + | \forall x \in \mathbf{R}: | ||
| + | 1 + x + \frac{x^2}{2!} + | ||
| + | \frac{x^3}{3!} + \cdots\\ | ||
| + | & | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | < | ||
| + | </ | ||
| + | $ \displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k^2} = \frac{\pi^2}{6}$ | ||
| + | |||
| + | ==== Integrale ==== | ||
| + | < | ||
| + | $ \int\limits_0^{\frac{\pi}{3}}\cos x~\mathrm{d}x \qquad$ | ||
| + | $ \int\int_D\limits f(x, y) \mathrm{d}x \mathrm{d}y \qquad | ||
| + | \int\!\!\!\int_D\limits f(x, y)\, | ||
| + | </ | ||
| + | $ \int\limits_{-\infty}^{\infty}\displaystyle{\frac{1}{1 + x^2}} \mathrm{d}x \qquad$ | ||
| + | $ \int\limits_0^{\frac{\pi}{3}}\cos x~\mathrm{d}x \qquad$ | ||
| + | $ \int\int_D\limits f(x, y) \mathrm{d}x \mathrm{d}y \qquad | ||
| + | \int\!\!\!\int_D\limits f(x, y)\, | ||
| + | |||
| + | $ \prod_{i=1}^n i = n! \qquad \prod\limits_{i=1}^n i = n! \qquad \prod\nolimits_{i=1}^n i = n!$ | ||
| + | $ \prod_{i=1}^n i = n! \qquad \prod\limits_{i=1}^n i = n! \qquad \prod\nolimits_{i=1}^n i = n!$ | ||
| + | |||
| + | $ \displaystyle{{n \choose k}} = \frac{\displaystyle\prod_{i=1}^n i} {\displaystyle\prod_{i=1}^k i\cdot \prod_{i=1}^{n-k} i}$ | ||
| + | $ \displaystyle{{n \choose k}} = \frac{\displaystyle\prod_{i=1}^n i} {\displaystyle\prod_{i=1}^k i\cdot \prod_{i=1}^{n-k} i}$ | ||
| + | ==== Funktionen ==== | ||
| + | $ \displaystyle \lim_{x \to 0} \frac{\sin x}{x}=1$ | ||
| + | $ \displaystyle \lim_{x \to 0} \frac{\sin x}{x}=1$ | ||
| + | $ \displaystyle \int \frac{\mathrm{d}x} {\sin a x \cos a x} = \frac{1}{a} \ln \tan a x$ | ||
| + | $ \displaystyle \int \frac{\mathrm{d}x} {\sin a x \cos a x} = \frac{1}{a} \ln \tan a x$ | ||
| + | $ \arcsin x = \left[ \arccos \sqrt{1 - x^2}\right]$ | ||
| + | $ \arcsin x = \left[ \arccos \sqrt{1 - x^2}\right]$ | ||
| + | ==== Matrizen ==== | ||
| + | < | ||
| + | a_{11} & a_{12} & \cdots & a_{1n} \\\\ | ||
| + | a_{21} & a_{22} & \cdots & a_{21} \\\\ | ||
| + | \vdots & \vdots & \ddots & \vdots \\\\ | ||
| + | a_{m1} & a_{m2} & \cdots & a_{mn} | ||
| + | \end{array}$</ | ||
| + | $ \begin{array}{|cccc|} | ||
| + | a_{11} & a_{12} & \cdots & a_{1n} \\\\ | ||
| + | a_{21} & a_{22} & \cdots & a_{21} \\\\ | ||
| + | \vdots & \vdots & \ddots & \vdots \\\\ | ||
| + | a_{m1} & a_{m2} & \cdots & a_{mn} | ||
| + | \end{array}$ | ||
| + | |||
| + | < | ||
| + | \Gamma_{11} & \Gamma_{12} & \cdots & \Gamma_{1n}\\\\ | ||
| + | \Gamma_{21} & \Gamma_{22} & \cdots & \Gamma_{2n}\\\\ | ||
| + | \vdots & \vdots & \ddots & \vdots\\\\ | ||
| + | \Gamma_{m1} & \Gamma_{m2} & \cdots & \Gamma_{mn} | ||
| + | \end{array}\right\}$ | ||
| + | </ | ||
| + | |||
| + | $ \left\{\begin{array}{cccc} | ||
| + | \Gamma_{11} & \Gamma_{12} & \cdots & \Gamma_{1n}\\\\ | ||
| + | \Gamma_{21} & \Gamma_{22} & \cdots & \Gamma_{2n}\\\\ | ||
| + | \vdots & \vdots & \ddots & \vdots\\\\ | ||
| + | \Gamma_{m1} & \Gamma_{m2} & \cdots & \Gamma_{mn} | ||
| + | \end{array}\right\}$ | ||
| + | |||
| + | < | ||
| + | x & \textrm{fuer } x \ge 0\\\\ | ||
| + | -x & \textrm{fuer } x < 0\\\\ | ||
| + | \end{array}\right\}$ | ||
| + | </ | ||
| + | |||
| + | $ |x|= \left\{ \begin{array}{ll} | ||
| + | x & \textrm{fuer } x \ge 0\\\\ | ||
| + | -x & \textrm{fuer } x < 0\\\\ | ||
| + | \end{array}\right\}$ | ||
| + | |||
| + | < | ||
| + | \begin{array}{c@{}c@{}c} | ||
| + | \begin{array}{|cc|} | ||
| + | \hline | ||
| + | a_{11} & a_{12} \\\\ | ||
| + | a_{21} & a_{22} \\\\ | ||
| + | \hline | ||
| + | \end{array} & 0 & 0 \\\\ | ||
| + | 0 & \begin{array}{|ccc|} | ||
| + | \hline | ||
| + | b_{11} & b_{12} & b_{13}\\\\ | ||
| + | b_{21} & b_{22} & b_{23}\\\\ | ||
| + | b_{31} & b_{32} & b_{33}\\\\ | ||
| + | \hline | ||
| + | \end{array} & 0 \\\\ | ||
| + | 0 & 0 & \begin{array}{|cc|} | ||
| + | \hline | ||
| + | c_{11} & c_{12} \\\\ | ||
| + | c_{21} & c_{22} \\\\ | ||
| + | \hline | ||
| + | \end{array} \\\\ | ||
| + | \end{array} | ||
| + | \right)$ | ||
| + | </ | ||
| + | |||
| + | $ \left( | ||
| + | \begin{array}{c@{}c@{}c} | ||
| + | \begin{array}{|cc|} | ||
| + | \hline | ||
| + | a_{11} & a_{12} \\\\ | ||
| + | a_{21} & a_{22} \\\\ | ||
| + | \hline | ||
| + | \end{array} & 0 & 0 \\\\ | ||
| + | 0 & \begin{array}{|ccc|} | ||
| + | \hline | ||
| + | b_{11} & b_{12} & b_{13}\\\\ | ||
| + | b_{21} & b_{22} & b_{23}\\\\ | ||
| + | b_{31} & b_{32} & b_{33}\\\\ | ||
| + | \hline | ||
| + | \end{array} & 0 \\\\ | ||
| + | 0 & 0 & \begin{array}{|cc|} | ||
| + | \hline | ||
| + | c_{11} & c_{12} \\\\ | ||
| + | c_{21} & c_{22} \\\\ | ||
| + | \hline | ||
| + | \end{array} \\\\ | ||
| + | \end{array} | ||
| + | \right)$ | ||
| + | < | ||
| + | </ | ||
